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On the Rotation Matrix in Minkowski Space-Time
Abstract
In this paper, a Rodrigues-like formula is derived for 4 × 4 semi skew-symmetric real matrices in E41. For this purpose, we use the decomposition of a semi skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi skew-symmetric matrices A1 and A2 satisfying the properties A1A2=0, A31=A1 and A32=-A2. Then, we find Lorentzian rotation matrices with semi skew-symmetric matrices by Rodrigues-like formula. Furthermore, we give a way to find the semi skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = eA.
... We simplify and expand our previous work [6] (a comment on [11]). Our propositions admit simple proofs that avoid specialized terminology or advanced theories. ...
... The matrix calculus, involving analytic functions f (A) with matrix argument, has been of great interest (cf., e.g., [11], [6], and references therein) since the origins of linear algebra, with the exponential e Z playing the leading role. ...
... On the other hand, a little calculation shows that tr (F 2 ) = 2(d 2 − h 2 ) and with (11) in mind we arrive at the stated equations. ...
Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a G-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.
... In general, split quaternions are used to express rotation and reflection transformations for timelike and spacelike vectors in Minkowski 3-space. Because the units of split quaternions are compatible with the unit base vectors in Minkowski 3-space [3], [12], [13], [15]. Rotations around the lightlike axis were also investigated in [4], [10]. ...
... Theorem 18 Let S be a skew symmetric matrix in the form (15) and v = xi+yj+zk is a quadratic vector and on the quadric surface Q. Then the matrix exponential ...
... Let S is a skew-symmetric matrix with v = xi+yj+zk in the form (15). If v is not a unit spacelike vector, then is a quadratic rotation matrix where v is the rotation matrix. ...
Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form. This article examines how to find the corresponding number system for any quadratic form. As a result of this examination, bilinear form, vector product, skew-symmetric matrix, and rotation matrices corresponding to any number system were obtained.
... This is illustrated in Sect. 2.3, where knowing the rotations in the Minkowski 3-space (from now on, referred to as hyperbolic rotations) is found to be quite essential (Özdemir and Erdogdu 2014). Moreover, the rigid movement can be determined by using the symmetries of the regular planar l-polygon and, in this way, we recover T completely. ...
... The invariance of (5)-(6) under hyperbolic rotations follows from the invariance of the Minkowski cross-product under them (Özdemir and Erdogdu 2014). Thus, given a hyperbolic rotation matrix R, such that R·T(s, 0) = T(s, 0) and R·X(s, 0) = X(s, 0), if the solution is unique, then R · X(s, t) = X(s, t), R · T(s, t) = T(s, t), for all t. ...
... First, let us mention that, at the level of the NLS equation, the hyperbolic case is not much different from the Euclidean case; however, the obtention of X and T depends entirely on hyperbolic rotations (Özdemir and Erdogdu 2014;Ratcliffe 2006). In this regard, following the approach in de la Hoz and Vega (2014), we observe that, by definition, ψ(s, 0) is l-periodic, and since (11) is invariant with respect to space translations, ψ(s, t) is also l-periodic, for all t ∈ R. On the other hand, ψ(s, 0) = e irks ψ(s, 0), r = 2π/l, l > 0; thus, from the Galilean invariances of (11), ...
The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.
... Thus, by integrating the generalized Frenet-Serret formulas at times that are rational multiples of l 2 /(2π), we obtain the evolution of the curve X and of the tangent vector T, up to a rigid movement. This has been illustrated in Section 2.3, where knowing the rotations in the Minkowski 3-space (from now on, referred to as hyperbolic rotations) is found to be quite essential [31]. Moreover, the rigid movement can be determined by using the symmetries of the regular planar l-polygon and, in this way, we recover T completely. ...
... 2.1.1. Spatial symmetries of X and T. The invariance of (5)-(6) under hyperbolic rotations follows from that of the Minkowski cross product under them [31]. Thus, given a hyperbolic rotation ...
... Problem formulation and the behavior at rational multiples of the time period. Let us first mention that at the level of the NLS equation, the hyperbolic case is not much different from the Euclidean case; however, the obtention of T and X depends entirely on hyperbolic rotations [31,5]. In this regard, following the approach in [18], we observe that, by definition, ψ(s, 0) is l-periodic, and, since (10) is invariant with respect to space translations, ψ(s, t) is also l-periodic for all t ∈ R. On the other hand, ψ(s, 0) = e irks ψ(s, 0), r = 2π/l, l > 0; thus, from the Galilean invariances of (10), ψ(s, t) = e irks−i(rk) 2 t ψ(s − 2rkt, t), for all k. ...
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
... With the appropriate selection of α and β in generalized space, rotational motions such as circular, hyperbolic, and elliptical can be obtained. Thus, this gives the opportunity to work in many areas, such as from the centrifugal force taken into account when calculating the slope of the road to the motion of an electric field [24] and the orbits of the planets [25,26]. Generalized space is also a scalar product space. ...
... Generalized space is also a scalar product space. For general information about scalar product spaces and their algebraic properties, see [16][17][18]20,[24][25][26][27][28][29]. ...
This paper aims to investigate the screw motion in generalized space. For this purpose, firstly, the change in the screw coordinates is analyzed according to the motion of the reference frames. Moreover, the special cases of this change, such as pure rotation and translation, are discussed. Matrix multiplication and the properties of dual numbers are used to obtain dual orthogonal matrices, which are used to simplify the manipulation of screw motion in generalized space. In addition, the dual angular velocity matrix is calculated and shows that the exponential of this matrix can represent the screw displacement in the generalized space. Finally, to support our results, we give two examples of screw motion, the rotation part of which is elliptical and hyperbolic.
... Among the linear transformations, orthogonal transformations are the well known ones. These transformations contribute to the solution of many problems in kinematics, physics and computer graphics [1][2][3][4][5][6][7][8]. There are different types of orthogonal transformations: reflections, rotations and their various combinations. ...
... In this study, We aim to investigate the second kind of orthogonal transformations i.e. rotations in four dimensional Euclidean space. In the literature, there are many studies dealing with rotation matrices in three dimensional space [10][11][12][13][14][15][16] while there are limited studies examining rotation matrices in higher dimensions [6][7][8][9]. Mostly, three dimensional rotation matrices have been analyzed with the help of skew symmetric matrices [14][15][16]. For any given skew-symmetric matrix with the use of the property A 3 = −A. ...
The main topic of this study is to investigate rotation matrices in four dimensional Euclidean space in two different ways. The first of these ways is Rodrigues formula and the second is Cayley formula.The most important common point of both formulas is the use of skew symmetric matrices. However, depending on the skew symmetric matrix used, it is possible to classify the rotation matrices by both formulas. Therefore, it is also revealed how the rotation matrices obtained by both formulas are classified as simple, doubleor isoclinic rotation. Eigenvalues of skew symmetric matrices play the major role in this classification. With the use of all results, it is also seen which skew symmetric matrix is obtained from a given rotation matrix by Rodrigues and Cayley formula, respectively. Finally, an algorithm for classification of rotations is given with the help of the obtained datas and explained with an example.
... do not seem to have made it into the textbook literature, although experts on the Lorentz group know that this property holds [10][11][12]. Riesz proved it using the decomposition of Minkowski space into different invariant subspaces under Lorentz transformations. Furthermore, exponentials of general so(1, 3) elements ...
... have been evaluated in closed form (generalized Rodrigues formulae) both in the vector and the spinor representations of the Lorentz group [11][12][13][14], and both Zeni and Rodrigues, as well as Özdemir and Erdoğdu, have pointed out that these formulae can in principle be used to answer the question of Lie algebra coverage for + affirmatively by comparing the closed forms of (16) with equation (8) and demonstrating that the ...
Physicists know that covering the continuously connected component + ↑ of the Lorentz group can be achieved through two Lie algebra exponentials, whereas one exponential is sufficient for compact symmetry groups like SU(N) or SO(N). On the other hand, both the general Baker-Campbell-Hausdorff formula for the combination of matrix exponentials in a series of higher order commutators, and the possibility to define the logarithm ln ( M ̲ ) of a general matrix M ̲ through the Jordan normal form, seem to naively suggest that even for non-compact groups a single exponential should be sufficient. We provide explicit constructions of ln ( M ̲ ) for all matrices M ̲ in the fundamental representations of the non-compact groups SL ( 2 , R ) , SL ( 2 , C ) , and SO(1, 2). The construction for SL ( 2 , C ) also yields logarithms for SO(1, 3) through the spinor representations. However, it is well known that single Lie algebra exponentials are not sufficient to cover the Lie groups SL ( 2 , R ) and SL ( 2 , C ) . Therefore we revisit the maximal neighbourhoods 1 ⊂ SL ( 2 , R ) and 1 , C ⊂ SL ( 2 , C ) which can be covered through single exponentials exp ( X ̲ ) with X ̲ ∈ sl ( 2 , R ) or X ̲ ∈ sl ( 2 , C ) , respectively, to clarify why ln ( M ̲ ) ∉ sl ( 2 , R ) or ln ( M ̲ ) ∉ sl ( 2 , C ) outside of the corresponding domains 1 or 1 , C . On the other hand, for the Lorentz groups SO(1, 2) and SO(1, 3), we confirm through construction of the logarithm ln ( Λ ̲ ) that every transformation Λ ̲ in the connectivity component + ↑ of the identity element can be represented in the form exp ( X ̲ ) with X ̲ ∈ so ( 1 , 2 ) or X ̲ ∈ so ( 1 , 3 ) , respectively. We also examine why the proper orthochronous Lorentz group can be covered by single Lie algebra exponentials, whereas this property does not hold for its covering group SL ( 2 , C ) : The logarithms ln ( Λ ̲ ) in + ↑ correspond to logarithms on the first sheet of the covering map SL ( 2 , C ) → + ↑ , which is contained in 1 , C . The special linear groups and the Lorentz group therefore provide instructive examples for different global behaviour of non-compact Lie groups under the exponential map.
... Therefore, the Bäcklund transformation has important role in soliton theory. For example; Bäcklund transformation and soliton equations for KP equation were investigated in [8] and modern applications of Backlund and Darboux transformations in soliton theory were deeply discussed in [9]. ...
... It is time to examine the rotation matrix with respect to kind of rotation plane. For rotation matrices in Minkowski space-time, the readers are referred to [9,18,19]. ...
Bu çalışmanın amacı, Minkowski
uzay-zamanda timelike eğriler arasında Bäcklund dönüşümünü tanımlamaktır. Bu
amaç doğrultusunda, timelike Bäcklund eğrilerin Frenet çatıları arasında
ilişkiyi ortaya koyan dönme matrisinin seçimine bağlı olarak dönüşümü
inceledik. İkisi spacelike hiperdüzlemde küresel dönme ve biri ise timelike
hiperdüzlemde hiperbolik dönme olmak üzere üç farklı dönme matrisi durumu söz
konusudur. Her durum için, timelike Bäcklund eğrilerinin eğrilik fonksiyonları
arasındaki ilişki ortaya konmuştur. Bu arada, işaret farkı gözeterek timelike Bäcklund
eğrilerin eşit ikinci burulma fonksiyonuna sahip olması gerektiği
ispatlanmıştır. Bu aynı zamanda; Bäcklund dönüşümün bir sabit ikinci burulmaya sahip
timelike eğriyi bir başka sabit ikinci burulmaya sahip timelike eğriye taşıyan
dönüşüm olduğu anlamıma gelir.
... Also, the rotation matrix about a null axis is studied in [7] by using the same methods. In R 1,3 , they were obtained by means of the corresponding decomposition in [11] and with Cayley formula in [4]. ...
... by using (11) and the last equation of (10). Since the g-vector product satisfies the relation (5), we can write ...
... We simplify and expand our previous work [6] (a comment on [10]). Our propositions admit simple proofs that avoid specialized terminology or advanced theories. ...
... The matrix calculus, involving analytic functions f (A) with matrix argument, has been of great interest (cf., e.g., [10], [6], and references therein) since the origins of linear algebra, with the exponential e Z playing the leading role. ...
Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a G-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.
... Among the linear transformations, orthogonal transformations are the well known ones. These transformations contribute to the solution of many problems in kinematics, physics and computer graphics [1][2][3][4][5][6][7][8]. There are different types of orthogonal transformations: reflections, rotations and their various combinations. ...
... In this study, We aim to investigate the second kind of orthogonal transformations i.e. rotations in four dimensional Euclidean space. In the literature, there are many studies dealing with rotation matrices in three dimensional space [10][11][12][13][14][15][16] while there are limited studies examining rotation matrices in higher dimensions [6][7][8][9]. Mostly, three dimensional rotation matrices have been analyzed with the help of skew symmetric matrices [14][15][16]. For any given skew-symmetric matrix with the use of the property A 3 = −A. ...
The main topic of this study is to investigate rotation matrices in four dimensional Euclidean space in two
different ways. The first of these ways is Rodrigues formula and the second is Cayley formula.The most
important common point of both formulas is the use of skew symmetric matrices. However, depending on
the skew symmetric matrix used, it is possible to classify the rotation matrices by both formulas. Therefore,
it is also revealed how the rotation matrices obtained by both formulas are classified as simple, double
or isoclinic rotation. Eigenvalues of skew symmetric matrices play the major role in this classification.
With the use of all results, it is also seen which skew symmetric matrix is obtained from a given rotation
matrix by Rodrigues and Cayley formula, respectively. Finally, an algorithm for classification of rotations
is given with the help of the obtained data and explained with an example.
... In this method, only three numbers are needed to construct a rotation matrix in the Euclidean 3-space ( [27], [28], [29] and, [30]).The vector set up with these three numbers gives the rotation axis. This method can be extended to the n dimensional Euclidean and Lorentzian spaces ( [34], [6], [5], [22] and, [15]). ...
... In the 4 dimensional Euclidean and Lorentzian spaces, a skew symmetric matrix is decomposed as A = θ 1 A 1 + θ 2 A 2 using two skewsymmetric matrices A 1 and A 2 satisfying the properties A 1 A 2 = 0, A 3 1 = −A 1 and A 3 2 = −A 2 . Hence, the Rodrigues and Cayley rotation formulas can be used to generate 4 dimensional rotation matrices ( [34], [3], [18], [4], and [15]). ...
Elliptical rotation is the motion of a point on an ellipse through some angle
about a vector. The purpose of this paper is to examine the generation of
elliptical rotations and to interpret the motion of a point on an elipsoid
using elliptic inner product and elliptic vector product. To generate an
elliptical rotation matrix, first we define an elliptical ortogonal matrix and
an elliptical skew symmetric matrix using the associated inner product. Then we
use elliptic versions of the famous Rodrigues, Cayley, and Householder methods
to construct an elliptical rotation matrix. Finally, we define elliptic
quaternions and generate an elliptical rotation matrix using those quaternions.
Each method is proven and is provided with several numerical examples.
... For the proof of Lemma 1, the readers are referred to [12]. ...
... Cayley formula in Minkowski space-time semi-skew-symmetric matrices which is given in [12]. If θ 2 = 0, then the Lorentzian rotation matrix can be found as ...
In this paper, Cayley formula is derived for 4 x 4 semi-skew-symmetric real matrices in E-1(4). For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = theta(1)A(1) + theta(2)A(2) by two unique semi-skew-symmetric matrices A(1) and A(2) satisfying the properties A(1)A(2) = 0, A(1)(3) = A(1) and A(2)(3) = -A(2). Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay(A).
... Rotation matrices have wide applications in many fields such as mathematics, physics, computer science, mechanic, kinematics, H. B. Çolakoğlu and M. Özdemir MJOM and geometry. Rotation matrices in the various spaces can be found in many studies [1][2][3][4][5][6][7][8][9]. The isometry group for the bilinear or sesquilinear forms can be found in the Macykey et al's paper [10]. ...
In this paper, we determine elliptical motions that occur on any given ellipsoid in 3D space without using affine transformations. To this end, first, we define the generalized Euclidean inner product whose sphere is the given ellipsoid, and determine skew symmetric matrices and the generalized vector product related to the 3D generalized Euclidean inner product space. Finally, we generate elliptical rotation matrices in 3D generalized Euclidean space using the famous Rodrigues, Cayley, and Householder methods. The formulas and results obtained are supported with numerical examples. We also give an algorithm for generalized elliptical rotation.
... In general, split quaternions are used to express rotation and reflection transformations for timelike and spacelike vectors in Minkowski 3-space because the units of split quaternions are compatible with the unit base vectors in Minkowski 3-space. [7][8][9][10][11][12] Rotations around the lightlike axis were also investigated in previous studies, 13,14 but in the study, frames consisting of one spacelike and two lightlike vectors were presented, and rotations around the lightlike axis were examined. While creating this frame, the scalar product of two lightlike vectors is taken as 1, but the scalar product of any two different lightlike vectors is not 1. ...
This article aims to define the rotational motion around a lightlike axis in pseudo‐null or null frames more easily and to put forward a suitable number system to express rotation transformation in these frames. Therefore, for null and pseudo‐null frames, we define a new set of numbers consisting of a linear combination of two lightlike units and a spacelike unit, which we call Cartan numbers. The first part gives the properties of the set of Cartan numbers, and in the second part, some geometric definitions, theorems, and applications are given. In addition, the definition of the center of a parabola and the concept of the central angle are defined. The connection is made between these concepts and the parabolic rotation transformation. Rotation matrices for parabolic rotations are obtained. The rotating points indicated that it follows in which cases a line path and in which cases a parabolic path. The rotation transformation along any straight line or parabola is studied with examples.
... But the orthonormal matrix varies according to the bilinear form and the metric of the space. For example, 3-and 4-dimensional rotation matrices in the Lorentzian space can be found in articles [1][2][3]. Rotation matrices form a special orthogonal group and are used extensively for geometry, kinematics, computer graphics, and animations involving the rigid body movements. In the 2-dimensional Euclidean and Lorentzian space, a rotation matrix can easily be generated using basic linear algebra, Rodrigues, Cayley and Householder formulas. ...
In this paper, we determine non-parabolic conical motions that occur on any given ellipse or hyperbola without using affine transformations. To achieve this aim, first, we define a generalized inner product whose circle is the given ellipse or hyperbola, and then determine elliptical and hyperbolic versions of skew-symmetric and orthogonal matrices using the associated inner product. Finally, we generate elliptical and hyperbolic versions of rotation and reflection matrices using the famous Rodrigues, Cayley, and Householder transformations. For each of the generalized formulas, we give numerical examples.
... Rotation transformations around timelike, spacelike, or null axis are defined very differently from one another. For further information about rotation transformations, see [20,21,22,23,24,25]. In two different situations, the pseudo null frames (⃗, ⃖ ⃗, ⃖ ⃗ ) and (⃗, ⃖⃗ , ⃖⃖ ⃗ ) are described with a linear transformation of one another. ...
Considering the importance of Minkowski space in physics, it is an incomplete approach to deal with EM waves only in Euclidean space. For this reason, this paper deals with EM waves along pseudo null curves in Minkowski space. The main purpose of this study is to examine electromagnetic waves by defining an adapted orthogonal frame along the EM wave which contains both electric and magnetic fields. For this purpose, the extended derivative formulas of pseudo null frame are obtained. Depending on the values of Bishop curvatures, the linear transformations between the pseudo null frame and EM wave vector fields are described in two cases. For all these cases, the relations between these frames are stated, respectively. Moreover, the derivative formulas EM wave vectors are stated by means of geometric phase. Furthermore, the necessary and sufficient conditions provided by the geometric phase are expressed for EM wave vectors to be parallel transportation of the pseudo null frame. Finally, an application is given to investigate the obtained results.
... Using (28) and expression for ψ θ (s, t pq ), we can write Ψ θ (s, t pq ) = (α k,m +iβ k,m )(s, t pq ) = ρ q e iζ k,m . As a result, integrating the generalized Frenet-Serret frame gives the rotation matrix R k,m (see [19, (35)]) which performs a rotation of time-like angle ρ q about a space-like rotation axis (0, − sin(ζ k,m ), cos(ζ k,m )) [34]. In other words, it describes the transition across a corner at s =s k where for the CHP,s k = (2π(k + 1)/M q + s pq ) − ; on the other hand, for the numerical computations we work with a truncated HHP with M sides, length L = lM , ands k = l(k + 1)/q + s pq , for k = 0, 1, . . . ...
The main purpose is to describe the evolution of \Xt = \Xs \wedge_- \Xss, with \X(s,0) a regular polygonal curve with a nonzero torsion in the 3-dimensional hyperbolic space. Unlike in the Euclidean space, a nonzero torsion implies two different helical curves. However, recent techniques developed by de la Hoz, Kumar, and Vega help us in describing the evolution at rational times both theoretically and numerically, and thus, the similarities and differences. Numerical experiments show that the trajectory of the point \X(0,t) exhibits new variants of Riemann's non-differentiable function whose structure depends on the initial torsion in the problem. As a result, with these new solutions, it is shown that the smooth solutions (helices, straight line) in the hyperbolic space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in coherence with some recent theoretical results obtained by Banica and Vega.
... Now, let the curve follow a hyperbolic path. For the further information about the Lorentz transformations used in this case, see [6]. ...
In this study, a model of accretive growth for arbitrary surfaces in three-dimensional Minkowski space is formulated by evolving a curve. An analytical approach to surfaces is also given in terms of a few parameters which are effective in the accretive growth of surfaces. The proposed method is visualized on some test surfaces and displayed in terms of figures.
... For example, These matrices about arbitrary lightlike axis in Minkowski 3space were studied in [13] by deriving the Rodrigues' rotation formula and using the corresponding Cayley map. In Euclidean 4-space and Minkowski space-time, they were obtained by means of the corresponding decomposition in [7,17]. On the other hand, the paper [16] examined the elliptical rotation which states the motion of a point on an ellipse or ellipsoid by defining the elliptical product space. ...
Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by , . In this paper, we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the Lorentzian Scalar Product Space such that the general hyperboloids are the pseudo-spheres of . We adapt the Rodrigues, Cayley, and Householder methods to and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix.
... Also, the rotation matrix about a null axis is studied in [7] by using the same methods. In R 1,3 , they were obtained by means of the corresponding decomposition in [11] and with Cayley formula in [3]. Rotations on different geometric shapes can be examined using an appropriate bilinear form. ...
In this paper the generalization of the rotations on any lightcone in Minkowski 3-space Rg1,2 is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix.
... For example, These matrices about arbitrary lightlike axis in Minkowski 3- space were studied in [10] by deriving the Rodrigues' rotation formula and using the corresponding Cayley map. In Euclidean 4-space and Minkowski space-time, they were obtained by means of the corresponding decomposition in [6] [14]. On the other hand, the study [13] examined the elliptical rotation which states the motion of a point on an ellipse or ellipsoid by de…ning the elliptical product space. ...
Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by a 1 x 2 + a 2 y 2 + a 3 z 2 = ; 2 R +. In this paper, we investigate the hyperbolic rotation matrices in order to get the motion of a point about a …xed point or axis on the general hyperboloids by de…ning the Lorentzian Scalar Product Space R 2;1 a 1 a 2 a 3 such that the general hyperboloids are the pseudo-spheres of this space. We adapt the Rodrigues, Cayley, and Householder methods to R 2;1 a 1 a 2 a 3 and de…ne hyperbolic split quaternions to obtain an hyperbolical rotation matrix.
... Exponentials of a real skew-symmetric matrices and logarithms of orthogonal matrices have many applications in kinematics [6], rigid body dynamics [8], geometry and physics, as well as in robotics [2], computer graphics [3], motion interpolations [7] and other areas [13]. In Euclidean 4-space and Minkowski space-time, rotation matrix is generated by using the corresponding decomposition in [3,11]. ...
We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues’ rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation Rq
about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples.
The present paper gives an extraordinary view of the normal congruence of surfaces including the s−lines and b−lines in terms of electromagnetic wave vectors in ordinary space. Frenet Serret frame of given a space curve are described in E3 in terms of anholonomic coordinates which includes eight parameters. Using the expression the Frenet frames and electromagnetic wave vectors on the curve with a linear transformation in terms of each other, the changes of t⃗, E⃗ and B⃗ between any two points in the tangential and binormal direction along with the curved path σ=σ(s,n,b) are obtained in terms of geometric phase β, respectively. Morover, the solution of the systems of differential equations of optical fiber with position vector is obtained. Intrinsic geometric properties of this normal congruence of surfaces are obtained in terms of electromagnetic wave vectors. The conditions under which electromagnetic and magnetic vectors satisfy Maxwell’s equations given electric charge and current densities are investigated. Finally, an application is stated to investigate a normal congruence of surfaces by using electromagnetic wave vectors. Also, we give an illustrations of polarization and magnetic field vector of EM wave.
In this study, the reflections in and are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case.
In this study, the reflections in í µí´¼ 3 and í µí´¼ 4 are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in í µí´¼ 3 with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in í µí´¼ 4 with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case. Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar ÖZET: Bu çalışmada, í µí´¼ 3 ve í µí´¼ 4 uzayında yansımalar birim kuaterniyonlar ile incelenmiştir. İlk olarak, í µí´¼ 3 uzayında orjinden geçen ve kuaterniyona dik doğrultudaki doğru boyunca yansımayı belirten bir lineer dönüşüm tanımlanmıştır. Ardından, ortaya çıkan sonuçlar örneklendirilmiştir. Benzer şekilde, í µí´¼ 4 uzayında orjinden geçen hiperdüzlem ve kuaterniyon doğrultusundaki doğru boyunca yansımalara karşılık gelen dönüşümler tanıtılmıştır. Son olarak bu yansıma dönüşümlerinin matris temsilleri elde 2 edilmiş ve her durum için bu özdeğer ve özvekörlerin hesaplanması ile geometrik yorumlar kuaterniyon katsayıları ile analiz edilmiştir.
In this study, the reflections in í µí´¼ 3 and í µí´¼ 4 are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in í µí´¼ 3 with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in í µí´¼ 4 with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case .
THIS PAPER is a part of the paper "An Alternative Approach to Elliptical Motion". Advances in Applied Clifford Algebras, 2015. Elliptical rotation is the motion of a point on an ellipse through some angle about a vector. The purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an elipsoid using elliptic quaternions, elliptic inner product and elliptic vector product. In this paper, we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions.
Elliptical rotation is the motion of a point on an ellipse through some angle about a vector. The purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an ellipsoid using elliptic inner product and elliptic vector product. To generate an elliptical rotation matrix, first we define an elliptical ortogonal matrix and an elliptical skew symmetric matrix using the associated inner product. Then we use elliptic versions of the famous Rodrigues, Cayley, and Householder methods to construct an elliptical rotation matrix. Finally, we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions. Each method is proven and is provided with several numerical examples.
This paper is being kept "on hold " in arxiv.org for over two months. The moderators are unable to "classify it" - that is the official reason given. Highly improbable. More probable is that something is going on behind the curtain. I do not think it is just incompetence ....
We comment on the article by M. Ozdemir and M. Erdogdu. We indicate that the exponential map onto the Lorentz group can be obtained in two elementary ways. The first way utilizes a commutative algebra involving a conjugate of a semi-skew-symmetric matrix, and the second way is based on the classical epimorphism from SL(2,C) onto SO_0(3,1)
In this paper the formula of the exponential matrix A e when A is a semi skew-symmetric real matrix of order 4 is derived. The formula is a generalization of the Rodrigues formula for skew-symmetric matrices of order 3 in Minkowski 3-space.
. In this study, by using Lorentzian matrix multiplication, Cayley formula and Euler parameters of a Lorentzian orthogonal matrix
are obtained in Lorentz space L3. Then, by using Euler parameters of a rotation in a split quaternion, the split quaternion equation of a rotation movement
is obtained in the space L3.
. We show that there is a generalization of Rodrigues's formula for computing the exponential map exp: so(n) ! SO(n) from skew symmetric matrices to orthogonal matrices when n 4, and we give a method for computing the function log: SO(n) ! so(n). The case where Gamma1 is an eigenvalue of R 2 SO(n) requires a special treatment. The key idea is the decomposition of a skew symmetric n Theta n matrix B in terms of skew symmetric matrices B 1 ; : : : ; Bm such that B = ` 1 B 1 + Delta Delta Delta + ` mBm B i B j = B j B i = 0 n (i 6= j); B 3 i = GammaB i ; where (i` 1 ; Gammai` 1 ; : : : ; i` m ; Gammai` m ) are the nonnull eigenvalues of B. We also consider the exponential map exp: se(n) ! SE(n), where se(n) is the Lie algebra of the Lie group SE(n) of (affine) rigid motions. We show that there is a Rodrigues-like formula for computing this exponential map, and we give a method for computing log: SE(n) ! se(n). This yields a direct proof of the surjectivity of exp: se(n) ! ...
The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised correspondingly. Inequalities involving the angles through which the coordinate axes in 3D space are displaced are used to prove that the left-hand and the right-hand quaternions are each other's inverses, thus proving the Euler-Rodrigues formula. A general procedure to determine the Euler parameters of a given 3D rotation matrix is sketched. By equating the leftmost top element to -1 instead of +1 in the general 4D rotation matrix, one proves the counterpart of the Euler-Rodrigues formula for 3D rotoreflections. Keywords: Euler--Rodrigues formula, Euler parameters, quaternions, four--dimensional rotations, three--dimensional rotations, rotoreflections
It is fairly well known that rotation in three dimensions can be expressed as
a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A
generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric
generating matrix is derived for the rotation matrix of tensors of order n.
The Euler-Rodrigues formula for rigid body rotation is recovered by n=1. A
Cayley form of the n-th order rotation tensor is also derived. The
representations simplify if there exists some underlying symmetry, as is the
case for elasticity tensors such as strain and the fourth order tensor of
elastic moduli. A new formula is presented for the transformation of elastic
moduli under rotation: as a 21-vector with a rotation matrix given by a
polynomial of degree 8. Explicit spectral representations are constructed from
three vectors: the axis of rotation and two orthogonal bivectors. The tensor
rotation formulae are related to Cartan decomposition of elastic moduli and
projection onto hexagonal symmetry.
With the aid of quaternion algebra, rotation in Euclidean space may be dealt with in a simple manner. In this paper, we show that a unit timelike quaternion represents a rotation in the Minkowski 3-space. Also, we express Lorentzian rotation matrix generated with a timelike quaternion.
In this short paper the formula of the exponential matrix e
A when A is a kew-symmetric real matrix of order 4 is derived. The formula is a generalization of the well known Rodrigues formula for skew-symmetric matrices of order 3.
Catadioptric Projective Geometry: Theory and Applications, Doctorial Dissertation,University of Pennsylvania
- C M Geyer
C. M. Geyer: Catadioptric Projective Geometry: Theory and Applications, Doctorial Dissertation,University of Pennsylvania 2003.
Norris: Euler–Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors
A. N. Norris: Euler–Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors, Math. Mech.
Solids 13 (2008): 465-498.
Notes on Differential Geometry and Lie Groups, University of Pennsylvania 2014
- J H Gallier
J. H. Gallier: Notes on Differential Geometry and Lie Groups, University of Pennsylvania 2014; 730 p.
Mebius: Derivation of Euler–Rodrigues formula for three-dimensional rotations from the general formula for four dimensional rotations, arxiv: math
J. E. Mebius: Derivation of Euler–Rodrigues formula for three-dimensional rotations from the general
formula for four dimensional rotations, arxiv: math.GM (2007).
- J H Gallier
J. H. Gallier: Geometric Methods and Applications, For Computer Science and Engineering, Texts Appl.
Math. 38 (2011): 680p.
Wilkens: Quaternions and Rotations in E 4
- J L Weiner
J. L. Weiner and G. R. Wilkens: Quaternions and Rotations in E 4, Am. Math. Mon. (2005), 69-76.